To find the product of \(\sqrt{7}\) and \(2\sqrt{2}\), we can follow the steps below.
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Write the expression for the product: \[ \sqrt{7} \cdot 2\sqrt{2} \]
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Rearrange the expression: \[ 2 \cdot \sqrt{7} \cdot \sqrt{2} \]
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Combine the square roots: Using the property of square roots \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), we get: \[ \sqrt{7} \cdot \sqrt{2} = \sqrt{14} \] So the expression becomes: \[ 2\sqrt{14} \]
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Simplify the product: The product is \(2\sqrt{14}\).
Now, we need to determine whether this result is rational or irrational.
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A rational number can be expressed as the quotient of two integers, whereas an irrational number cannot be expressed in this form.
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\(\sqrt{14}\) is an irrational number because 14 is not a perfect square. Therefore, \(2\sqrt{14}\), which is a non-zero multiple of an irrational number, is also irrational.
Conclusion: The product of \(\sqrt{7}\) and \(2\sqrt{2}\) in simplest form is \(2\sqrt{14}\), and this result is irrational.